DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§2. The Gauss Map and the Second Fundamental Form 5318. Suppose U ⊂ R 3 is open and x: U → R 3 is a smooth map (of rank 3) so that x u , x v , and x ware always orthogonal. Then the level surfaces u = const, v = const, w = const form a triplyorthogonal system of surfaces.a. Show that the spherical coordinate mapping x(u, v, w) =(u sin v cos w, u sin v sin w, u cos v)furnishes an example.b. Prove that the curves of intersection of any pair of surfaces from different systems (e.g.,v = const and w = const) are lines of curvature in each of the respective surfaces. (Hint:Differentiate the various equations x u · x v =0,x v · x w =0,x u · x w =0with respect to themissing variable. What are the shape operators of the various surfaces?)19. In this exercise we analyze the surfaces of revolution that are minimal. It will be convenient towork with a meridian as a graph (y = h(u), z = u), as opposed to our customary parametrizationof surfaces of revolution.a. Use Exercise 1.2.4 and Proposition 2.5 to show that the principal curvatures arek 1 =h ′′(1 + h ′2 ) 3/2 and k 2 = − 1 h · 1√ .1+h ′2b. Deduce that H =0if and only if h(u)h ′′ (u) =1+h ′ (u) 2 .c. Solve the differential equation. (Hint: Either substitute z(u) =lnh(u)orintroduce w(u) =h ′ (u), find dw/dh, and integrate by separating variables.) You should find that h(u) =1ccosh(cu + b) for some constants b and c.20. By choosing coordinates in R 3 appropriately, we may arrange that P is the origin, the tangentplane T P M is the xy-plane, and the x- and y-axes are in the principal directions at P .a. Show that in these coordinates M is locally the graph z = f(x, y) = 1 2 (k 1x 2 + k 2 y 2 )+...,...where limx,y→0 x 2 + y 2 =0.(Youmay start with Taylor’s Theorem: if f is C2 ,wehavewheref(x, y) =f(0, 0) + f x (0, 0)x + f y (0, 0)y +limx,y→0...x 2 + y 2 = 0.)12(fxx (0, 0)x 2 +2f xy (0, 0)xy + f yy (0, 0)y 2) + ...,b. Show that if P is an elliptic point, then a neighborhood of P in M ∩ T P M is just theorigin itself. Show that if P is a hyperbolic point, such a neighborhood is a curve thatcrosses itself at P and whose tangent directions at P are the asymptotic directions. Whathappens in the case of a parabolic point?21. Let P ∈ M be a non-planar point, and if K ≥ 0, choose the unit normal so that l, n ≥ 0.a. We define the Dupin indicatrix to be the conic in T P M defined by the equation II P (V, V) =1. Show that if P is an elliptic point, the Dupin indicatrix is an ellipse; if P is a hyperbolicpoint, the Dupin indicatrix is a hyperbola; and if P is a parabolic point, the Dupinindicatrix is a pair of parallel lines.b. Show that if P is a hyperbolic point, the asymptotes of the Dupin indicatrix are given byII P (V, V) =0, i.e., the set of asymptotic directions.